Proto Dual Algebroid Cohomology,

The main work of this thesis is to generalize the Dirac theories from Lie bialge-broid to the protobialgebroid. The protobialgebroid is depicted by the language of supermanifold. Some basic properties of protobialgebroid are discussed in this thesis in detail. We study the Dirac structures on protobialgebroid, give the integrable conditions for maximally isotropic subbundle being Dirac structure for protobialgebroid by the notion of characteristic pair. From the integrable conditions, we found out that Dirac structure has closed relations with twisting of protobialgebroid. Further, the closed relations are explained, the fact that twisting of protobialgebroids is a equivalence relation is verified. We define two kinds of reducible Dirac structures, discuss the problem on the reduction of protobialgebroids. As a simple application of the theories of Dirac structure on protobialgebroid, we concretely discussed the reduction of twisted Poisson manifold.Lie quasi-bialgebroid is a special case of protobialgebroid, we discuss some of its properties in detail, define the exact Lie quasi-bialgebroid and the triangular Lie quasi-bialgebroid, give the particularities of corresponding Dirac structures, discuss the relations of the triangular Lie quasi-bialgebroid with the quasi-YangBaxter equation.The algebra language(big bracket) and the method of treating the protobialgebroid are all very valuable. In the end, Nijenhuis tensor is introduced into the theories of protobialgebroid, making the Nijenhuis structures have more abundant research contents. From the supermanifold point of view, Each relation of objects on Nijenhuis structure is very clearly simple and directly. We discuss compatibilities of deform brackets, deform operators, differential operators in the Nijenhuis structures. As a simple application of the homological method, we define and study G-J differential in the theories of Jcaobi bialgebroid, give the definition and the example of generalized Lie quasi-bialgebroid.